Properties

Label 2-3450-5.4-c1-0-21
Degree $2$
Conductor $3450$
Sign $0.894 - 0.447i$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − 2·11-s i·12-s + 4i·13-s + 16-s − 6i·17-s + i·18-s + 8·19-s + 2i·22-s i·23-s − 24-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s + 1.10i·13-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 1.83·19-s + 0.426i·22-s − 0.208i·23-s − 0.204·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3450} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.482737436\)
\(L(\frac12)\) \(\approx\) \(1.482737436\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 \)
23 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.108792932344571634927519014254, −7.906007519820990042421048372561, −7.34789011723112045438592156963, −6.33198267606459058781549834071, −5.20999647729956396555254722645, −4.90337421142677275128985205570, −3.87611049359390698065887618396, −3.07629251146014194041135800566, −2.27947879961208929765099071462, −0.945099906505614831662834854500, 0.56485119263180306817068232408, 1.80728213568130872542643341006, 3.08154558190132171322615153754, 3.79431825527151714091635283865, 5.10195536559024912931784383956, 5.58876043468952255389389342209, 6.23059551650532439949833130639, 7.29623657180821848906325632604, 7.64364110372990049015430630430, 8.323516246670792470470583458685

Graph of the $Z$-function along the critical line